Invertible spectra in the $E(n)$-local stable homotopy category
Mark Hovey and Hal Sadofsky
Recall that the Picard group, first introduced into stable homotopy
theory by Hopkins, is the group of isomorphism classes of
smash-invertible spectra. For the ordinary stable homotopy category,
this group is just Z on the spheres S^n. For localized
categories,however the situation may be more complex. Hopkins,
Mahowald, and Sadofsky, and also Strickland, have studied the Picard
group of the K(n)-local category. In this paper we study the Picard
group of the L_n-local, or E(n)-local, stable homotopy category. We
find that if n is large relative to the prime p, the answer is just Z
again.
The proof involves a few general results that should be of independant
interest. In particular, we give two proofs of a generalized
Miller-Ravenel change of rings theorem, one of which depends on a
general (unfortuately unpublished, but not tremendously difficult)
change of rings theorem due to Hopkins. We also give an E(n) version of
the Landweber filtration theorem, and show that the E(n) Adams spectral
sequence always converges.
We conclude the paper with the simplest example where n is not large
relative to p, when n=1 and p=2. Here the Picard group is Z plus Z/2.